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Toy models, Tadashi Tokieda | LMS Popular Lectures 2008

London Mathematical Society・2 minutes read

Two jars with different fill levels are rolled down an inclined plane, showcasing varied speeds due to moment of inertia, with experiments showing how the descent speed depends on grain amount and the angle of repose, leading to surprising phenomena in mechanics of contact and illustrating conservation principles through various toys and experiments. The analysis delves into factors like friction, angular momentum, and stability, culminating in determining the time for a toy to tip over and highlighting the mysterious behavior of spinning toys like the chiral tippy-top that defy traditional explanations, presenting an open problem for mechanics experts.

Insights

  • The experiments with jars on an inclined plane demonstrate how the speed of descent varies based on the amount of grains inside, showcasing the impact of moment of inertia on motion.
  • The analysis of spinning toys reveals intricate relationships between symmetry, mass distribution, and conservation laws, leading to perpetual spinning behavior governed by energy conservation and unique velocity vector relationships.

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Recent questions

  • How does the speed of descent vary in jars with different grain levels?

    The speed of descent in jars with varying grain levels was observed through two experiments. In Experiment 1, one jar was filled 2/3 full and the other 1/3 full, resulting in the 2/3 full jar descending slowly with grains sliding down, while the 1/3 full jar stopped midway down the slope. This variation in speed was attributed to the moment of inertia and the amount of grains inside the jars affecting the center of gravity and torque exerted by gravity.

  • What is the significance of symmetry breaking in mechanics of contact?

    Symmetry breaking in the mechanics of contact leads to unexpected phenomena, as seen in various toys and experiments like the penguin toy and collision experiments. This phenomenon highlights surprising mechanics involving friction, rubbing, and viscous fluids, showcasing how symmetry breaking can result in unique and unpredictable outcomes in the movement of objects.

  • How is momentum conservation demonstrated in experiments with a penguin toy?

    Momentum conservation is illustrated through experiments with a penguin toy guarding a magnetic ball. The recoil of the system absorbs momentum, balancing out the movement, with more balls coming back at slower speeds to compensate for the momentum. This demonstration showcases the fundamental principle of momentum conservation in the movement of objects.

  • What role does the coefficient of friction play in analyzing stability?

    The coefficient of friction, denoted as mu, measures the difficulty of one material sliding over another and is crucial in analyzing the stability of an object. By calculating the angle at which sliding begins and considering variables like mu, mg, friction, M, R, and Omega, the rate of change of the angle of inclination, denoted as theta dot, can be determined to assess stability.

  • How does mass distribution affect the spinning direction of a toy?

    Mass distribution plays a crucial role in determining the spinning direction of a toy, particularly in chiral dynamics. The distribution of mass must be skewed with respect to the axis of symmetry for the toy to exhibit chiral behavior. Vibrations like pitching and rolling can influence the spinning direction, with pitching causing forward motion and rolling leading to backward motion, showcasing the intricate relationship between mass distribution and spinning behavior in toys.

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Summary

00:00

"Grains vs. Air: Slope Speed Experiments"

  • An individual with an inclined plane presents two jars, one filled with top-quality basmati rice from India and the other with top-quality air from Birmingham.
  • The jars are rolled down the slope, showing different speeds due to the moment of inertia.
  • The investigation focuses on how the speed of descent depends on the amount of grains inside the jars.
  • Experiment 1: Filling one jar 2/3 full and the other 1/3 full to observe the speed of descent.
  • Experiment 1 results in the 2/3 full jar going down very slowly with an avalanche of grains sliding down the surface.
  • Experiment 2: Filling one jar 1/3 full to observe the speed of descent.
  • Experiment 2 results in the jar stopping completely dead midway down the slope.
  • The angle of repose, denoted by alpha, explains the phenomenon of grains' descent based on the center of gravity and torque exerted by gravity.
  • The experiments highlight surprising phenomena in the mechanics of contact, involving friction, rubbing, and viscous fluids.
  • Symmetry breaking in mechanics of contact leads to unexpected phenomena, illustrated through various toys and experiments like the penguin toy and collision experiments.

18:44

Magnetic Momentum: Neodymium Experiments and Conservation

  • Potential energy converts to kinetic energy when an object rolls down a precipice.
  • Momentum conservation is crucial to understanding the movement of objects.
  • A penguin guarding a magnetic ball demonstrates momentum conservation.
  • The recoil of a system absorbs momentum, balancing out the movement.
  • More balls coming back at slower speeds compensate for the momentum.
  • Neodymium magnets are highly magnetizable and used in unique experiments.
  • The largest hardware store in Tokyo stocks neodymium magnets.
  • Shooting a magnet from the opposite side reveals a unique magnetic phenomenon.
  • A magnetic abacus demonstrates a mysterious and unexplained phenomenon.
  • Conservation of angular momentum is illustrated through a toy called billboquet.

35:44

"Friction, Stability, and Tipping: Analyzing Dynamics"

  • Friction is determined by the coefficient of friction, denoted as mu, which measures the difficulty of one material sliding over another.
  • The coefficient of friction can be calculated by measuring the angle at which sliding begins, typically less than 45 degrees.
  • The rate of change of the angle of inclination, denoted as theta dot, is crucial in analyzing the stability of an object.
  • To determine theta dot, a combination of variables like mu, mg, friction, M, R, and Omega must be considered.
  • The presence of a proportionality constant in the analysis affects the final result, with a constant close to one being common.
  • Calculating the time for an object to tip over involves dividing pi by theta dot, with various numerical values like radius, gravitational acceleration, and angular spin being essential.
  • The estimated time for tipping over a toy is around 2.4 seconds, based on the given numerical values and calculations.
  • The toy's behavior is related to chiral dynamics, where symmetry and mass distribution play a crucial role in determining the direction of spin.
  • The distribution of mass must be skewed with respect to the axis of symmetry for the toy to exhibit chiral behavior.
  • Vibrations like pitching and rolling can influence the toy's spinning direction, with pitching causing forward motion and rolling leading to backward motion.

52:26

"Spinning Toy Physics: Energy Conservation and Mystery"

  • The world record for a spinning toy was held by a former Cavendish professor, Brian Peapod, who carved a wine bottle to create a spinning toy that reversed up to seven times.
  • The analysis of the spinning toy involves a model with three motions denoted as P (pitching), R (rolling), and S (spin), leading to a simple equation involving frequency ratios and vector products.
  • The equation derived from the analysis reveals two constants of motion, indicating conservation of energy and a unique relationship between velocity vectors.
  • The spinning toy exhibits periodic motion due to the conservation laws, with solutions forming closed curves and the toy continuing to spin indefinitely with minimal dissipation.
  • An open problem is presented with a chiral tippy-top toy that defies explanation, challenging mechanics experts and highlighting the unpredictable nature of scientific phenomena.
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